Terminating circuit in a battery feed circuit of an electronic exchange system

Abstract

 Disclosed is a terminating circuit in a battery feed circuit of an electronic exchange system. The battery feed circuit comprises two transconductance amplifiers connected by a coupling capacitor, for supplying a direct current through a ring terminal and a tip terminal to a load. Each of the transconductance amplifiers comprises an operational amplifier (21); a resistor (R_F) connected between the output end of the operational amplifier and the tip terminal or the ring terminal, and an impedance unit connected between the inverting input terminal of the operational amplifier and the output end. The impedance unit has an impedance determined in such a way that the required terminating impedance of the battery feed circuit when viewed from the tip terminal and the ring terminal is formed only by the internal elements including the impedance means and the resistor, whereby an external larger-size capacitance capacitor for matching becomes unnecessary.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

[0001] The present invention relates to a terminating circuit in a battery feed circuit of an electronic exchange system, and more particularly, to a circuit for realizing the terminating impedance of a battery feed circuit, which supplies a direct current to subscriber lines or to trunk lines, by an impedance within a transconductance amplifier.

2. Description of the Related Art

[0002] The transconductance amplifier is described in, for example, the "Operational Amplifier Circuit Design Handbook" in the MAGROWHILL Electronic Circuit Technology Series, written by Staut and Caufman, or in the "IEEE Journal of Solid State Circuits, Vol.SC-16, No. 4, August, 1981 in "A High-Voltage IC for a Transformerless Trunk and Subscriber Line Interface, by Denis W. Aull etc.

[0003] A prior art battery feed circuit is disclosed in U.S. Patent No. 4,677,669, issued on June 30, 1987.

[0004] To supply a current to subscriber lines or to transit trunk lines in an electronic exchange system, a battery feed circuit must be constructed which has a characteristic having a resistance of about 200 Ω for a direct current and having a high impedance for a differential alternating current, to have an impedance matching the load connected to the subscriber lines or to the transit trunk lines.

[0005] A terminating impedance of the lines for an alternating current, from the load side, is generally determined as, for example, 600 Ω + 2 µF or 900 Ω + 2 µF. Conventionally, the terminating impedance is connected between the subscriber lines, i.e., between a tip line terminal and a ring line terminal. The 2 µF capacitor should have the tolerance voltage of -48 V applied between the subscriber lines, and further, should have a high precision capacity satisfying a return loss regulation. Accordingly, a problem arises in the conventional terminating circuit in that the size of the capacitor is large, and the presence of this large size capacitor disadvantageously enlarges the size of the battery feed circuit as a whole.

SUMMARY OF THE INVENTION

[0006] An object of the present invention is to provide a terminating circuit of a battery feed circuit having a stable terminating impedance, without using large-size parts, by appropriately setting an impedance connected to an operational amplifier in a transconductance amplifier in the battery feed circuit so that the impedance is equal to the terminating impedance of the lines when viewed from the output side of the operational amplifier.

[0007] To obtain the above object, according to the present invention, there is provided a terminating circuit in a battery feed circuit of an electronic exchange system. The battery feed circuit comprises two transconductance amplifiers connected by a coupling capacitor for supplying a direct current through a ring terminal and a tip terminal to a load. Each of the transconductance amplifiers comprises: an operational amplifier; a resistor connected between the output end of the operational amplifier and the tip terminal or the ring terminal; and an impedance unit connected between the inverting input terminal of the operational amplifier and the output end.

[0008] The impedance unit has an impedance determined in such a way that the required terminating impedance of the battery feed circuit when viewed from the tip terminal and the ring terminal is formed only by the internal elements including the impedance unit and the resistor.

[0009] According to one aspect of the present invention, the impedance unit comprises a resistor and a capacitor connected in series.

[0010] According to another aspect of the present invention, the impedance unit comprises only a resistor.

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The above object and features of the present invention will be more apparent from the following description of the preferred embodiments with reference to the drawings, wherein:

Figure 1 is a circuit diagram of a conventional subscriber line circuit for explaining the background of the present invention;

Fig. 2 is a circuit diagram showing a conventional terminating circuit of a battery feed circuit;

Fig. 3A is a diagram explaining the equivalent circuit of a capacitor when the circuit shown in Fig. 2 is divided into two;

Fig. 3B is a diagram explaining the equivalent circuit of a series connected resistor and a capacitor when the circuit shown in Fig. 2 is divided into two;

Fig. 3C is the equivalent circuit diagram of half of the circuit shown in Fig. 2;

Fig. 4 is a circuit diagram for explaining the principle of the present invention;

Fig. 5 is a circuit diagram of a terminating circuit according to a first embodiment of the present invention;

Fig. 6 is a circuit diagram of a terminating circuit according to a second embodiment of the present invention; and

Fig. 7 is an equivalent circuit diagram of the circuit shown in Fig. 6.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0012] For a better understanding of the present invention, a conventional circuit will be first described.

[0013] Figure 1 is a circuit diagram of a conventional subscriber line circuit, for explaining the background of the present invention. In the figure, the subscriber line circuit has the BORSHT functions, i.e., battery feed (B), overvoltage protection (O), ringing (R), supervision (S), hybrid (H), and testing (T) functions. The present invention relates to the battery feed circuit and an improvement of the battery feed (B) function.

[0014] Figure 2 is a circuit diagram showing an example of a terminating circuit in a conventional battery feed circuit. In the figure, 11 and 12 are transconductance amplifiers; 21 and 22 are operational amplifiers; 31 and 32 are feedback resistors; 41 and 42 are resistors connected to the outputs of the operational ampli­fiers 21 and 22; 51 and 52 and 61 and 62 are voltage dividing resistors, i.e., gain resistors; 71 and 72 are resistors for connecting the transconductance ampli­fiers 11 and 12; and 8 is a capacitor for cutting a direct current between the transconductance ampli­fiers 11 and 12.

[0015] A resistor 9 and a capacitor 10 are connected in series between a ring line terminal (A) 501 and a tip line terminal (B) 502, to provide a terminating impedance. The resistances of the resistors 31 and 32 are expressed as R_f1; the resistances of the resistors 41 and 42 are expressed as R_F; the resistances of the resistors 51 and 52 are expressed as R_f2; the resistances of the resistors 61 and 62 are expressed as R_S2; the resistances of the resistors 71 and 72 are expressed as R_S1; the capacitance of the capacitor 8 is expressed as C_AB; and the resistance of the resistor 9 and the capacitance of the capacitor 10 for the terminating impedance are expressed, respectively, as R and C.

[0016] In the illustrated circuit, it is assumed that the resistances R_f1 , R_f2 , R_S1 , and R_S2 are the same, namely, R_f1 = R_f2 = R_S1 = R_S2. In the following description, both the resistors and the resistances may be expressed by the resistance values, and both the capacitors and the capacitances may be expressed by the capacitance values.

[0017] As will be apparent from the figure, the transconductance amplifiers 11 and 12 are symmetric with respect to the coupling capacitor 8.

[0018] When the circuit shown in Fig. 2 is divided into two along the dash dot line ℓ, the equivalent circuit of the coupling capacitor (C_AB)8 is expressed as a series connected two capacitors 2C_AB as shown in Fig. 3A, and the equivalent circuit of the series connected resistor 9 and the capacitor 10 are expressed as shown in Fig. 3B. Therefore, the equivalent circuit of the transconductance amplifier 11 and a half of the resistor 9 is as illustrated in Fig. 3C.

[0019] For the sake of easy explanation, the description is given for the equivalent circuit shown in Fig. 3C.

[0020] In Figs. 2 and 3C, the same reference numerals and symbols represent the same parts. The coupling capacitor 81 has a capacitance twice that of the coupling capacitor 8 in Fig. 2; the resistor 91 has a resistance R/2 half that of the resistor 9 in Fig. 2; and the capacitor 101 has a capacitance 2C twice that of the capacitor 10 in Fig. 2. In the equivalent circuit shown in Fig. 3, the input impedance Z_in when viewed from the tip line terminal (A) 501 is obtained as follows.

[0021] First, when an input voltage V_in is applied to the tip line terminal (A) 501, the voltage V_OP+ at the non-inverting input terminal of the operational amplifier 22 is divided by the resistors R_f2 and R_S2. The voltage at the inverting input terminal V_OP- of the operational amplifier 22 is expressed as V_OP+ = V_OP- because of the imaginary short of the operational amplifier 21. Further, when the impedance C_AB of the coupling capacitor 8 is determined to be negligible in comparison with the resistance R_s1 , the output voltage V_OP of the operational amplifier 21 can be calculated as follows:

[0022] From the expressions (1) and (2),

[0023] Here, assuming that R_f1 = R_f2 = R_s1 = R_s2 or R_f1/R_f2 = R_s1/R_s2 , then, V_OP = V_in is obtained.

[0024] In the equivalent circuit shown in Fig. 3, the voltage at the left hand side of the resistor R_F is V_in and the voltage at the right hand side of the resistor R_F is V_OP , which is equal to V_in. Therefore, the voltage across the resistor R_F is zero, and thus no current flows through the resistor R_F. Accordingly, the input impedance of the right side circuit when viewed from the resistor R_F is infinite. In conclusion, the impedance of the equivalent circuit when viewed from the tip line terminal 501 is R_f2 + R_s2. The resistance of the circuit for a direct current is expressed as:

[0025] For the operational amplifier 12 shown in the upper side in Fig. 2, a similar calculation as shown above can be applied. Accordingly, the input impedance Z_in between the tip line terminal 501 and the ring line terminal 502 is (R + 1/jωC).

[0026] Therefore, the input impedance Z_in of the battery feed circuit shown in Fig. 2 is formed by the impedance of the resistor 9 and the capacitor 10. The value of the input impedance is determined to be 600 Ω + 2 µF or 900 Ω + 2 µF as mentioned before, to match the load, and since the capacitance 2 µF is a large capacitance, it is realized by a large size part.

[0027] When the load is not a subscriber line but is a transmission line such as a trunkline and the like, the input impedance Z_in should be substantially a pure resistance R. In the voice band between 200 Hz to 3.4 KHz, the resistance R should satisfy the condition:

[0028] To satisfy the above condition, the capacitance C should be more than 10 µF. In this case, the capacitor becomes larger in size and it becomes difficult to mount it on the circuit board of the battery feed circuit.

[0029] In the above-described conventional circuit, the resistor 9 and the capacitor 10 are connected in series between the tip line terminal 501 and the ring line terminal 502. The capacitor 10 should have a high tolerance voltage, a large capacitance, and a high precision, and therefore, the capacitor must be large, and thus a problem arises of a large circuit scale of the battery feed circuit as a whole.

[0030] Accordingly, an object of the present invention is to provide a terminating circuit of a battery feed circuit having a stable terminating impedance without using large-size parts, by appropriately setting an impedance connected to an operational amplifier in a transconductance amplifier in the battery feed circuit so that the impedance is equal to the terminating impedance of the lines when viewed from the output side of the operational amplifier.

[0031] Namely, in the above described conventional circuit, the resistances R_f1 , R_s1 , R_f2 , and R_s2 are determined in such a way that the input impedance of the battery feed circuit constructed by the operational amplifiers is infinite. Nevertheless, by appropriately adjusting the resistances to bring the input impedance to the desired value, the conventionally connected resistor 9 and the capacitor 10 can be deleted.

[0032] Figure 4 is a circuit diagram for explaining the principle of the present invention. In the figure, 11a is a transconductance amplifier connected to a tip line terminal 501 or a ring line terminal 502; 21 is an operational amplifier in the transconductance ampli­fier 11a; 41 is a reference resistor, connected between the output of the operational amplifier 21 and the tip line terminal 501 or the ring line terminal 502, for determining the input impedance of the transconductance amplifier 11a; and 51 and 52 are voltage-dividing resistors. According to the present invention, the impedance unit 1 is connected between the inverting input terminal and the output terminal of the operational amplifier 21 so that the terminating impedance of the battery feed circuit when viewed from the tip line terminal and the ring line terminal is formed by only the reference resistor 41 and the impedance unit 1.

[0033] As explained in more detail in the following description of the embodiments, since the terminating impedance of the battery feed circuit is determined by only the reference resistor 41 and the impedance unit 1, the conventionally required large capacitance capacitor is not needed. The desired input impedance can be set by setting the voltage V_OP at the right hand side of the resistor R_F to a specific value by using the impedance unit 1, instead of setting the voltage V_in at the left hand side of the resistor R_F and the voltage V_OP at the right hand side of the resistor R_F to be equal in the conventional circuit.

[0034] Figure 5 is a circuit diagram of a terminating circuit according to a first embodiment of the present invention. In the figure, the same parts as those in the conventional circuit shown in Fig. 2 are denoted by the same reference numbers. In each of the trans­conductance amplifiers 11a and 12a, the difference from the conventional circuit is that a series circuit including a resistor (R_x) 201 and a capacitor (C_x) 202 connected in series is connected in parallel with the feedback resistor 31 or 32, subscriber terminal 82a is connected between the tip line terminal 501 and the ring line terminal 502, and a hybrid circuit 55 is connected between the tip line terminal 501 and the ring line terminal 502. A CODEC 56 is connected to the hybrid circuit 55.

[0035] When a differential signal of a speech sound is input between the tip line terminal 501 and the ring line terminal 502, the input impedance of the battery feed circuit is calculated as follows.

[0036] As an assumption, the resistances and the capacitances of the corresponding parts in the upper and lower circuits are the same. Namely, the resis­tances R_f1 , R_f2 , R_s1 , R_s2 , R_x and the capacitance C_x in the upper and lower transconductance amplifiers are assumed to be respectively the same.

[0037] When a differential signal is input between the tip line terminal 501 and the ring line terminal 502, i.e., between the lines A and B, the voltage between an end of the capacitor C_AB and the ground does not change. Therefore, for an alternating current, the potentials of the ends of the capacitor C_AB and the potential of the ground G are the same. Since the upper and the lower circuits are symmetric, the following analysis is applied to the lower transconductance amplifier 11a in the circuit shown in Fig. 5. The lower transconductance amplifier 11a is substantially equivalent to the principle circuit shown in Fig. 4.

[0038] The voltage V_OP⁺ at the non-inverting input of the operational amplifier 21 is expressed as:

[0039] The output voltage V_OP of the operational amplifier 21 is expressed as:

Assuming that the resistances R_f2 and R_s2 have a high resistance, then the current flowing through the resistors R_f2 and R_s2 can be neglected, and thus the input current I_in flowing through the resistor R_F expressed as:

[0040] The impedance Z of the battery feed circuit is:

[0041] Assuming that R_f2 = R_s2 = R_s1 , then the impedance Z is:

[0042] The impedance Z_u is obtained from the series circuit consisting of the resistor R_x and the capacitor C_x and the resistor R_f1 connected in parallel with the series circuit. Therefore, the resistance Z_u is expressed as:

[0043] Assuming that R_f1 = R_s1 , then the terminating impedance Z of the battery feed circuit is expressed as:

[0044] If a complex terminating impedance of, for example, 600 Ω + 2.16 µF, is required, the impedance required in one of the transconductance amplifiers is 300 Ω + 4.32 µF.

[0045] As an example, if R_F = 50 Ω and R_f1 = R_s1 = R_f2 = R_s2 = 100 KΩ, then the resistance R_x is 200 KΩ and the capacitance C_x is 4320PF.

[0046] As apparent from the above, description, the large capacitance of 2 µF is not necessary in the above embodiment, i.e., only a small capacitance of 4320PF is required, and accordingly, the complex terminating circuit can be stably realized by a small capacitance capacitor. The capacitance C_x is very small in comparison with the large capacitance of 2 µF necessary in the conventional circuit, and therefore, the circuit shown in Fig. 5 enables a miniaturizing of the battery feed circuit.

[0047] Figure 6 is a circuit diagram of a terminating circuit according to a second embodiment of the present invention, In the second embodiment, the explanation is given for when the terminating impedance of the battery feed circuit is a pure resistance R as required in transmission lines such as trunked lines. In the circuit also, the same parts as in Fig. 2 are denoted by the same reference numerals. The differences from the circuit in Fig. 2 are that the resistor and the capacitor for the terminating impedance are not connected between the tip line terminal 501 and the ring line terminal 502, and the feedback resistors (R′) 31a and 32a are connected in stead of the feedback resistors R_f1.

[0048] By this constitution also, the terminating impedance necessary for the battery feed circuit can be realized without using a large capacitance capacitor. In this case, the output voltage of the operational amplifier is explained in the following with reference to an equivalent circuit shown in Fig. 7.

[0049] Namely, to set the input impedance Z_in to be a pure resistance R/2, the potential V_B at the connecting point between the resistor R_F and the resistor R′ is made equal to the potential V_OP, when it is assumed that R = R_F + R. In Fig. 7, it will be seen that the following equation is established.

[0050] In Fig. 6 also, the upper circuit and the lower circuit are symmetric with respect to the capacitor C_AB. Therefore, the equivalent circuit of the transcon­ductance amplifier is the same as the circuit shown in Fig. 4.

[0051] When the input voltage V_in is input, the non-­inverting input voltage V_OP+ is a voltage divided value by the resistors R_f2 and R_s2. The input voltages V_OP+ and V_OP- become the same due to the imaginary short of the operational amplifiers.

[0052] When the impedance of the capacitance C_AB is determined to be negligible in comparison with the resistance R_s1 , the output voltage V_OP or the operational amplifier is calculated as follows:

[0053] Here, assuming that R_s1 = R_s2 = R_f2 = R_s , then the output voltage V_OP is:

[0054] To realize V_OP = V_B , it is sufficient to satisfy the relationship:

[0055] Further, the resistance R_f1 is:

[0056] Accordingly, by determining the resistance R_f1 as above, the resistor R and the capacitor C can be deleted from the conventional circuit, but even when the resistor and the capacitor are deleted, the battery feed characteristic is the same as in the conventional circuit.

[0057] In conclusion, in the circuit construction shown in Fig. 6, by setting the input impedance of the half line circuit to be R/2, the resistor R and the capacitor C, which are connected outside of the battery feed circuit, can be deleted. In particular, when the terminating impedance is a pure resistance, the capacitance C necessary in the conventional circuit is very large but according to the second embodiment of the present invention, the capacitor C is not necessary. Still further, according to the second embodiment of the present invention, since the capacitor C used in the conventional circuit is not used, the terminating resistance can be realized by a pure resistance.

[0058] As described above, according to the present invention, by adjusting the feedback resistance of an operational amplifier included in a transconductance amplifier to an appropriate impedance, the input impedance of the transconductance amplifier is made a finite value, whereby the conventionally required large capacitance capacitor for the terminating impedance becomes unnecessary, and as a result, the circuit scale of the battery feed circuit in an electronic exchange system can be largely reduced.

[0059] Reference signs in the claims are intended for better understanding and do not limit the scope.

Claims

1. A terminating circuit in a battery feed circuit of an electronic exchange system, said battery feed circuit comprising two transconductance amplifiers connected by a coupling capacitor (8) for supplying a direct current through a ring terminal and a tip terminal to a load, each of said transconductance amplifiers comprising:
an operational amplifier (21, 22);
a resistor (R_F) connected between the output end of said operational amplifier and said tip terminal or said ring terminal; and
impedance means (1a, 1b, 31a, 32a) connected between the inverting input terminal of said operational amplifier and said output end;
said impedance means having an impedance determined in such a way that the required terminating impedance of said battery feed circuit when viewed from said tip terminal and said ring terminal is formed only by the internal elements including said impedance means and said resistor.

2. A terminating circuit as claimed in claim 1, wherein said impedance means comprises a resistor and a capacitor connected in series.

3. A terminating circuit as claimed in claim 2, wherein each of said transconductance amplifier further comprises:
a feedback resistor (R_f1) connected between the inverting input terminal of said operational amplifier and the output end of said operational amplifier;
a first voltage-dividing resistor (R_f2) connected between the noninverting input terminal of said operational amplifier and said tip terminal of said operational amplifier and said tip terminal or said ring terminal;
a second voltage-dividing resistor (R_s2) connected between the noninverting input terminal of said operational amplifier and a power supply (V_A-BIAS , V_B-BIAS); and
a connecting resistor (R_S1) connected between the inverting input terminal of said operational amplifier and an end of said coupling capacitor.

4. A terminating circuit as claimed in claim 3, wherein said transconductance amplifiers are symmetric to each other so that the resistances of said feedback resistor, said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor in one of said transconductance amplifier are respectively the same as those in another one of said transconductance amplifiers.

5. A terminating circuit as claimed in claim 4, wherein the ends of said coupling capacitor can be deemed as ground for an alternating input signal applied between said ring terminal and said tip terminal.

6. A terminating circuit as claimed in claim 5, wherein when the resistances (R_f1 , R_f2 , R_s2 , R_s1) of said feedback resistor, said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor are respectively expressed as R_f1 , R_f2 , R_s2 , and R_s1 , the voltage at the non-­inverting input terminal of one of said operational amplifiers is expressed as:

7. A terminating circuit as claimed in claim 6, wherein when the resistances (R_f1 , R_f2 , R_s2 , R_s1) of said feedback resistor, said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor are respectively expressed as R_f1 , R_f2 , R_s2 , and R_s1 , and the parallel impedance of said impedance means and said feedback resistance is expressed as Zu, the output voltage V_OP of one of said operational amplifiers is expressed as:

8. A terminating circuit as claimed in claim 7, wherein the current through said first and second voltage dividing resistors R_f2 and R_s2 is negligible so that the input current I_in flowing through said resistor R_F is expressed as:

9. A terminating circuit as claimed in claim 8, wherein the input impedance Z of said battery feed circuit is expressed as:

10. A terminating circuit as claimed in claim 9, wherein when the resistances (R_f1 , R_f2 , R_s2 , R_s1) of said feedback resistor, said first voltage-dividing resistor, said second voltage-dividing resistor, and said connecting resistor are the same to each other, the input impedance Z of said battery feed circuit is expressed as:

where R_x is a resistance component in said impedance means and C_x is a capacitance component in said impedance means.

11. A terminating circuit as claimed in claim 1, wherein said impedance means comprises only a feedback resistor (R′).

12. A terminating circuit as claimed in claim 3, wherein each of said transconductance amplifiers further comprises:
a first voltage-dividing resistor (R_f2) connected between the noninverting input terminal of said operational amplifier and said tip terminal or said ring terminal;
a second voltage-dividing resistor (R_s2) connected between the noninverting input terminal of said operational amplifier and a power supply (V_A-BIAS , V_B-BIAS); and
a connecting resistor (R_s1) connected between the inverting input terminal of said operational amplifier and an end of said coupling capacitor.

13. A terminating circuit as claimed in claim 12 wherein said transconductance amplifiers are symmetric to each other so that the resistances of said feedback resistor, said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor in one of said transconductance amplifiers are respectively the same as those in another one of said transconductance amplifiers.

14. A terminating circuit as claimed in claim 13, wherein the ends of said coupling capacitor can be deemed as ground for an alternating input signal applied between said ring terminal and said tip terminal.

15. A terminating circuit as claimed in claim 14 wherein when the resistances (R_f2 , R_s2 , R_s1) of said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor are respectively expressed as R_f2 , R_s2 , and R_s1 , the voltage at the noninverting input terminal of one of said operational amplifiers is expressed as:

16. A terminating circuit as claimed in claim 15, wherein when the resistances (R′, R_f2 , R_s2 , R_s1) of said feedback resistor, said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor are respectively expressed as R′, R_f2 , R_s2 , and R_s1 , the output voltage V_OP of one of said operational amplifiers is expressed as:

17. A terminating circuit as claimed in claim 16, wherein when the resistances (R_f2 , R_s2 , R_s1) of said first voltage-dividing resistor, said second voltage dividing resistor, and said connecting resistor are the same as a resistance R_s , and when the resistance of said feedback resistance is expressed as R′ , then the output voltage V_OP of said operational amplifier is expressed as:

wherein V_in is the input voltage applied between said tip terminal or said ring terminal and ground.

18. A terminating circuit as claimed in claim 17, wherein when the required terminating impedance is expressed as R, the input voltage is expressed as V_in , and the resistance of said resistor (R_F) connected between the output end of said operational amplifier and said tip terminal or said ring terminal is expressed as R_F , then the voltage V_B at the output end of said operational amplifier is expressed as:

19. A terminating circuit as claimed in claim 18, wherein the resistance R′ of said feedback resistor is expressed as:

Drawing